Probabilistic analysis of continuous welded rail stability.

Ungureanu, Valentin-Vasile ; Dosa, Adam ; Botis, Marius Florin 等

1. INTRODUCTION

In 1992 / 1999 period the International Union of Railways (UIC) commissioned a research program from European Rail Research Institute (ERRI) about improving the knowledge of continuous welded rail (CWR) track, including switches (***, 1999). This research was necessary for revision and update of Leaflet UIC 720 which regulate the problems concerning the laying and maintenance of CWR track, which was from January 1986. In the new Leaflet UIC 720 (***, 2005), which was from March 2005, there were introduced concepts and criteria for the CWR buckling safety assessment and were shown case studies which appeal to the two CWR stability analysis softwares, one developed at TU Delft (Holland) for ERRI--software called initially CWERRI, and nowadays LONGSTAB--and the other developed at Foster&Miller company for Federal Rail Administration of United States of America (FRA)--software called CWR-BUCKLE (Kish & Samavedam, 1999). In this context, at the Civil Engineering Faculty from the University Transilvania of Brasov, Romania, was developed a software for track stability simulation using a non-linear discrete model for CWR buckling analysis, in the presence of thermal and vehicle loads, model called SCFJ (Stabilitatea Caii Fara Joante = Stability of CWR). A presentation of SCFJ model can be found in (Ungureanu, 2007).

This paper presents a probabilistic computational model of the buckling of CWR track. The great variability of the main parameters which characterize the stability of the track is introduced in the computational model by the statistical distribution of the parameters. The model is based on a nonlinear analysis in total Lagrangean formulation (Dosa & Ungureanu, 2007). The validity of the present model is verified through a series of comparative analyses with other author's results (Van, 1997).

In contrast with CWR-BUCKLE program, which use probability density functions, the probabilistic computational algorithm of SCFJ program is based on the evaluation of convolution integrals (Ghiocel & Lungu, 1975) in a discrete approach (Ghiocel & Lungu, 1982), using the histograms of the main parameters which characterize the stability of the CWR track (Ungureanu & Dosa, 2007).

2. ON SAFETY LIMITS

For every set of date which contain the physical and geometrical parameters of the CWR track introduced in the SCFJ model it will result (Fig. 1) a buckling response curve of track (***, 1999).

[FIGURE 1 OMITTED]

This curve is characterized by two points (Van, 1997):

[T.sub.b,max]--the maximum increase of temperature for which the buckling certainly starts, and

[T.sub.b,min]--the minimum increase of temperature which occurs in the post-buckling domain.

For a railway track safety conditions, [T.sub.allow] is the maximum allowable temperature above the neutral temperature of the rail that is considered safe as far as track buckling is concerned.

The safety concepts and criteria proposed by researchers are based on one of the following situations (***, 1999):

1. evaluation of [T.sub.b,max];

2. evaluating of [T.sub.b,min];

3. simultaneous quantification of [T.sub.b,max] and [T.sub.b,min].

Since the first situation leads to imprudent results in terms of safety, and the second method leads to too conservative results, it appears that the third method is the most rational, and therefore it is based on safety criteria developed by UIC.

For this reason the criterion of safety implemented in SCFJ program is provided by the new Leaflet UIC 720 through the ERRI D 202 Specialists' Committee (***, 2005):

[T.sub.R--TN] - [T.sub.N] [less than or equal to] [T.sub.allow] (1)

where [T.sub.R] is temperature of rail at a specific moment, [T.sub.N] is neutral temperature of rail and the [T.sub.allow] is computed as follows:

a) If [DELTA]T > 20[degrees]C:

[T.sub.allow] = [T.sub.b,min] + 0,25[DELTA]T (2)

b) If 5[degrees]C < [DELTA]T < 20[degrees]C:

[T.sub.allow] = [T.sub.b,min] (3)

c) If 0[degrees]C < [DELTA]T < 5[degrees]C:

[T.sub.allow] = [T.sub.b,min] - 5[degrees]C (4)

d) If [DELTA]T < 0[degrees]C: it is not allowable in main lines, where [DELTA]T = [T.sub.b,max] - [T.sub.b,min].

3. THE BUCKLING PROBABILITY EVALUATION

The essential aim of structural design is to ensure that in all sections the "minimum" sectional strengths are at least equal to the "maximum" structural effects of the loads (Ghiocel & Lungu, 1975).

The buckling "load" can be expressed in terms of the rail temperature increase over the neutral, and the "strength" is expressed in terms of the allowable temperature increase, [DELTA][T.sub.all-]

The buckling evaluation can be approached in a deterministic or in a probabilistic manner (Ungureanu, 2007).

In a deterministic approach the above criterion for buckling safety is satisfied or not. Hence, the track will either buckle out or not, and the "probability" of buckling is either 1 or 0.

In a probabilistic approach, the probability of load exceeding the strength is the "failure probability of the structure" and it can be evaluated on the basis of the so-called "convolution" integral (Ghiocel & Lungu, 1982) given below.

If [f.sub.R] (x) and [F.sub.R] (x) are the probability density function and the cumulative density function of the random sectional resistance, R, and, similarly, [f.sub.s] (x) and [F.sub.s] (x) are the probability density function and the cumulative distribution function of the random sectional load effect, S, the probability of failure is (Ghiocel & Lungu, 1975):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)

or

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6)

The integrals of equations (5) and (6) are called integrals of convolutions and they are solved by using the following relations for discrete distributions (Ghiocel & Lungu, 1982):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)

or

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8)

The input data of the SCFJ program are the histograms of key parameters of CWR track stability and the output data are the histogram of allowable temperature [T.sub.allow] and the histogram of difference between temperature of the rail at a specific moment and the neutral temperature of the rail ([T.sub.R] - [T.sub.N]). Then these results are introduced in the expressions of convolution integral to obtain the buckling probability versus rail temperature (Ungureanu, 2007).

4. CONCLUSION

1) CWR buckling under vehicle and thermal loads can be predicted using deterministic and probabilistic approaches. The deterministic approach will decide whether the CWR track with given parameters will buckle out or not. If it does not buckle, the "safety assurance" in terms of a buckling margin of safety can also be evaluated. The probabilistic approach introduces the statistical variability in the input parameters. For given statistical distributions of these parameters, the probabilistic approach gives the probability of buckling as a function of anticipated maximum rail temperature.

2) The probabilistic approach developed for CWR track buckling evaluations provides more flexibility in the maintenance of CWR tracks. Tradeoffs are possible between ballast lateral resistance, CWR neutral temperature and other parameters for more cost-effective maintenance for the same level of buckling risk.

3) A computational procedure for the determination of buckling probabilities has been formalized into a comprehensive buckling safety analysis program called SCFJ. The program incorporates both the deterministic and probabilistic analysis modules.

4) The SCFJ probabilistic method presented here can provide a rational basis for speed reductions for buckling risk mitigation when the rail temperature is above a "critical temperature". Allowable speed levels can also be determined using the method.

5) The main application of this probabilistic computational model is to assessment of temporary train speed limits using the simulation of the CWR track buckling in a probabilistic approach. The temporary train speed limits disturb normal passenger and freight traffic set in train schedule and determine losses due to the decrease of circulation capacity on the railway. An estimation of the allowable temperature limit under which it is possible to circulate in safety conditions with a known speed limit on the railway sector studied is an imperious necessity. In view of the great variability of main parameters which govern the stability of the CWR track it must use the algorithm for probabilistic assessment of the CWR track buckling developed in (Kish & Samavedam, 1999) and (Ungureanu, 2007) to estimate these temporary train speed limits.

5. REFERENCES

Dosa, A. & Ungureanu, V.V. (2007). Discrete model for the stability of continuous welded rail, In: Intersections/Intersectii, Vol.4, No.1, 2007, "Transportation Infrastructure Engineering", pp 25-34, ISSN 1582-3024

Ghiocel, D. & Lungu, D. (1975). Wind, Snow and Temperature Effects on Structures Based on Probability, Abacus Press, ISBN 0 85626 026 6, Tunbridge Wells, Kent

Ghiocel, D. & Lungu, D. (1982) Metode probabilistice in calculul constructiilor (Probabilistic methods in computation of structures), Editura Tehnica, Bucuresti

Kish, A. & Samavedam, G. (1999). Risk Analysis Based CWR Track Buckling Safety Evaluations, Available from: http://www.volpe.dot.gov/sdd/docs/idart-1299.pdf Accessed: 2009-07-29

Ungureanu, V.V. (2007). Cercetari privind simularea pierderii stabilitatii caii fara joante (Researches about simulation of continuous welded rail buckling), Teza de doctorat, Universitatea "Transilvania" din Brasov, Brasov, Romania.

Ungureanu, V.V. & Dosa, A. (2007). Algoritm pentru determinarea probabilitatii de pierdere a stabilitatii cadrului sine-traverse (Algorithm for computation of the rail track buckling probability), Lucrarile Sesiunii Stiintifice Constructii-Instalatii CIB 2007, 15-16 Noiembrie, Brasov, Romania, Talposi et al. (Ed.), pp 385-392, Editura Universitatii Transilvania, ISSN 1843-6617, Brasov, Romania

Van, M.A. (1997). Stability of Continuous Welded Rail Track, PhD Thesis, Delft University Press, ISBN: 90-407-1485-1, Delft, Netherlands

*** (1999) ERRID202-RP12 Improved knowledge of forces in CWR track (including switches)--Final report, European Rail Research Institute, Utrecht, Netherlands

*** (2005) UIC Leaflet -UIC 720R, Laying and Maintenance of CWR Track, 2nd edition, International Union of Railway, ISBN 2-7461-0526-8, Paris